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A026776 a(n) = Sum_{k=0..n} T(n,k), T given by A026769.

%I A026776 #7 Nov 01 2019 22:23:22
%S A026776 1,2,4,9,19,43,93,212,466,1070,2382,5506,12386,28800,65356,152745,
%T A026776 349183,819639,1885361,4441719,10270279,24269629,56363319,133529869,
%U A026776 311255601,738947515,1727873793,4109314729,9634406661,22946573863
%N A026776 a(n) = Sum_{k=0..n} T(n,k), T given by A026769.
%H A026776 G. C. Greubel, <a href="/A026776/b026776.txt">Table of n, a(n) for n = 0..1000</a>
%p A026776 T:= proc(n,k) option remember;
%p A026776    if n<0 then 0;
%p A026776    elif k=0 or k=n then 1;
%p A026776    elif n=2 and k=1 then 2;
%p A026776    elif k <= (n-1)/2 then
%p A026776         procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
%p A026776    else
%p A026776        procname(n-1,k-1)+procname(n-1,k) ;
%p A026776    end if ;
%p A026776 end proc;
%p A026776 seq(add(T(n, k), k=0..n), n=0..30); # _G. C. Greubel_, Nov 01 2019
%t A026776 T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[n==2 && k==1, 2, If[k<=(n-1)/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]]]; Table[Sum[T[n,k], {k,0,n}], {n,0,30}] (* _G. C. Greubel_, Nov 01 2019 *)
%o A026776 (Sage)
%o A026776 @CachedFunction
%o A026776 def T(n, k):
%o A026776     if (k==0 or k==n): return 1
%o A026776     elif (n==2 and k==1): return 2
%o A026776     elif (k<=(n-1)/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
%o A026776     else: return T(n-1,k-1) + T(n-1,k)
%o A026776 [sum(T(n,k) for k in (0..n)) for n in (0..30)] # _G. C. Greubel_, Nov 01 2019
%Y A026776 Cf. A026769, A026770, A026771, A026772, A026773, A026774, A026775, A026777, A026778, A026779.
%K A026776 nonn
%O A026776 0,2
%A A026776 _Clark Kimberling_